Problem: You have found the following ages (in years) of 4 gorillas. Those gorillas were randomly selected from the 20 gorillas at your local zoo: $ 7,\enspace 10,\enspace 15,\enspace 21$ Based on your sample, what is the average age of the gorillas? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 20 gorillas, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\overline{x}} = \dfrac{7 + 10 + 15 + 21}{{4}} = {13.3\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {39.69} + {10.89} + {2.89} + {59.29}} {{4 - 1}} $ {s^2} = \dfrac{{112.76}}{{3}} = {37.59\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{37.59\text{ years}^2}} = {6.1\text{ years}} $ We can estimate that the average gorilla at the zoo is 13.3 years old. There is also a standard deviation of 6.1 years.